Problem: Simplify the following expression: $\dfrac{72x^2}{108x^5}$ You can assume $x \neq 0$.
$ \dfrac{72x^2}{108x^5} = \dfrac{72}{108} \cdot \dfrac{x^2}{x^5} $ To simplify $\frac{72}{108}$ , find the greatest common factor (GCD) of $72$ and $108$ $72 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3$ $108 = 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3$ $ \mbox{GCD}(72, 108) = 2 \cdot 2 \cdot 3 \cdot 3 = 36 $ $ \dfrac{72}{108} \cdot \dfrac{x^2}{x^5} = \dfrac{36 \cdot 2}{36 \cdot 3} \cdot \dfrac{x^2}{x^5} $ $\phantom{ \dfrac{72}{108} \cdot \dfrac{2}{5}} = \dfrac{2}{3} \cdot \dfrac{x^2}{x^5} $ $ \dfrac{x^2}{x^5} = \dfrac{x \cdot x}{x \cdot x \cdot x \cdot x \cdot x} = \dfrac{1}{x^3} $ $ \dfrac{2}{3} \cdot \dfrac{1}{x^3} = \dfrac{2}{3x^3} $